How can Wall's theorem be generalised to non-simply connected manifolds? In a sense, this is a follow-up to this question.
By work of Freedman and Wall, it is known that if two simply-connected 4-manifolds $M$ and $N$ are homeomorphic, then there is $k \in \mathbb{N}$ such that $M \times \#^k (S^2 \times S^2)$ is diffeomorphic to $N \times \#^k (S^2 \times S^2)$.
(By $\#^k$, we denote $k$-fold connected sum.)
Can this statement be generalised to non-simply-connected manifolds, possibly including Whitehead torsion?
 A: For orientable 4-manifolds, it was shown by Gompf as Jeff says.  For compact nonorientable 4-manifolds with universal cover not spin, homeomorphic also implies stably diffeomorphic. 
On the other hand, Kreck showed that for every finitely presented group G with a surjective homomorphism $w \colon G \to C_2$, there is a nonorientable 4-manifold $M$ with 1-type $(G,w)$, with the following property.  The 4-manifold ($M$ plus the K3 surface) is homeomorphic to ($M$ plus eleven copies of $S^2 \times S^2$), but these two manifolds do not become diffeomorphic after adding copies of $S^2 \times S^2$.
https://link.springer.com/content/pdf/10.1007/BFb0075570.pdf
Thus there is a sense in which Wall's theorem fails quite badly to generalise to nonorientable 4-manifolds.
Here 1-type means (fundamental group, orientation character).
A: Gompf showed that the statement can indeed be generalized in this 1984 paper.  He proved that any two smooth structures on a compact 4-manifold become diffeomorphic after taking the connected sum with some number of copies of $S^2\times S^2$.
